The $n$-way number partitioning problem is a classic problem in combinatorial optimization, with applications to diverse settings such as fair allocation and machine scheduling. All these problems are NP-hard, but various approximation algorithms are known. We consider three closely related kinds of approximations. The first two variants optimize the partition such that: in the first variant some fixed number $s$ of items can be \emph{split} between two or more bins and in the second variant we allow at most a fixed number $t$ of \emph{splittings}. The third variant is a decision problem: the largest bin sum must be within a pre-specified interval, parameterized by a fixed rational number $u$ times the largest item size. When the number of bins $n$ is unbounded, we show that every variant is strongly {\sf NP}-complete. When the number of bins $n$ is fixed, the running time depends on the fixed parameters $s,t,u$. For each variant, we give a complete picture of its running time. For $n=2$, the running time is easy to identify. Our main results consider any fixed integer $n \geq 3$. Using a two-way polynomial-time reduction between the first and the third variant, we show that $n$-way number-partitioning with $s$ split items can be solved in polynomial time if $s \geq n-2$, and it is {\sf NP}-complete otherwise. Also, $n$-way number-partitioning with $t$ splittings can be solved in polynomial time if $t \geq n-1$, and it is {\sf NP}-complete otherwise. Finally, we show that the third variant can be solved in polynomial time if $u \geq (n-2)/n$, and it is {\sf NP}-complete otherwise. Our positive results for the optimization problems consider both min-max and max-min versions. Using the same reduction, and we provide a fully polynomial-time approximation scheme for the case where the number of split items is lower than $n-2$.

翻译：n路数字分割问题是组合优化中的经典问题，在公平分配和机器调度等多种场景中具有应用。这些问题均属于NP难问题，但存在多种近似算法。我们研究了三种密切相关的近似类型。前两种变体通过以下方式优化分割：第一种变体允许固定数量s的物品被分割到两个或更多容器中；第二种变体最多允许固定数量t的分割操作。第三种变体是决策问题：最大容器和必须落在预设区间内，该区间由固定有理数u乘以最大物品尺寸作为参数化边界。当容器数量n无界时，我们证明所有变体都是强NP完全的。当容器数量n固定时，运行时间取决于固定参数s、t、u。针对每个变体，我们完整刻画了其运行时间特征。对于n=2的情况，运行时间易于确定。我们的主要结果针对任意固定整数n≥3。通过第一种与第三种变体之间的双向多项式时间归约，我们证明：当s≥n-2时，带s个可分割物品的n路数字分割问题可在多项式时间内求解，否则为NP完全问题；当t≥n-1时，带t次分割操作的n路数字分割问题可在多项式时间内求解，否则为NP完全问题。最后，我们证明当u≥(n-2)/n时第三种变体可在多项式时间内求解，否则为NP完全问题。我们对优化问题的正向结果同时考虑了最小化最大值和最大化最小值两种版本。基于相同归约方法，我们为可分割物品数量小于n-2的情形提供了完全多项式时间近似方案。